Problem 36 - Differential Equations and Linear Algebra

Question

Use the Wronskian to show that the given functions are linearly independent on the given interval \(I\). $$\begin{aligned} f_{1}(x) &=\left\\{\begin{array}{cl} x^{2}, & \text { if } x \geq 0 ,\\\ 3 x^{3}, & \text { if } x<0, \end{array}\right.\\\ f_{2}(x)=7 x^{2}, I=(-\infty, \infty). \end{aligned}$$

Step-by-Step Solution

Answer

To show that the given functions \(f_1(x)\) and \(f_2(x)\) are linearly independent on the interval \(I=(-\infty, \infty)\) using the Wronskian method, we first find their derivatives: \(f'_{1}(x) = \left\\{\begin{array}{cl} 2x, & \text { if } x \geq 0 ,\\ 9x^{2}, & \text { if } x<0, \end{array}\right.\) and \(f'_{2}(x) = 14x\). We then compute the Wronskian for \(x \geq 0\) and \(x < 0\), which results in \(W(f_{1}, f_{2}) = 0\) for \(x \geq 0\) and \(W(f_{1}, f_{2}) = -21x^5\) for \(x < 0\). Since the Wronskian is not identically zero on the whole interval \(I=(-\infty, \infty)\), the given functions are linearly independent.

Step 1: Find the derivatives of the functions

Since the function \(f_1(x)\) is a piece-wise defined function, we need to find its derivative separately for \(x \geq 0\) and \(x < 0\). For \(x \geq 0\), we have: $$ f_{1}(x)=x^2 \implies f'_{1}(x) = 2x $$ For \(x<0\), we have: $$ f_{1}(x)=3x^3 \implies f'_{1}(x)=9x^2 $$ For the second function, \(f_2(x)\), find its derivative as follows: $$ f_{2}(x) = 7x^2 \implies f'_{2}(x) = 14x $$

Step 2: Compute the Wronskian

The Wronskian of the two given functions, denoted as \(W(f_{1}, f_{2})\), is calculated as follows: For \(x \geq 0\), $$ \begin{aligned} W(f_{1}, f_{2}) &= \begin{vmatrix} f_1(x) & f_2(x) \\ f'_1(x) & f'_2(x) \end{vmatrix}\\ &=\begin{vmatrix} x^2 & 7x^2 \\ 2x & 14x \end{vmatrix}\\ &= 14x^3 - 14x^3 = 0 \end{aligned} $$ For \(x < 0\), $$ \begin{aligned} W(f_{1}, f_{2}) &= \begin{vmatrix} 3x^3 & 7x^2 \\ 9x^2 & 14x \end{vmatrix}\\ &= 42x^5 - 63x^5 = -21x^5 \end{aligned} $$

Step 3: Determine if the Wronskian is non-zero

The Wronskian has two cases: - For \(x \geq 0\), \(W(f_{1}, f_{2}) = 0\). - For \(x < 0\), \(W(f_{1}, f_{2}) = -21x^5\). Since \(-\infty < x < 0\), \(-21x^5\) will be non-zero. Thus, on the whole interval \(I=(-\infty, \infty)\), the Wronskian is not identically zero. Therefore, the given functions are linearly independent.

Key Concepts

Linearly Independent

In the context of functions, linear independence means that no function in a set can be written as a combination of others in the same set. For two functions \(f_1(x)\) and \(f_2(x)\), they are linearly independent if there are no constants \(c_1\) and \(c_2\), not both zero, such that \(c_1 f_1(x) + c_2 f_2(x) = 0\) for all values of \(x\).

The Wronskian is a useful tool to test for linear independence of functions. If the Wronskian of these functions is non-zero at some point in the interval, the functions are linearly independent. Conversely, if the Wronskian is zero for all points in the interval, the functions might be dependent. So, we calculate the Wronskian to see where it is non-zero, indicating linear independence. This approach is crucial when you deal with scenarios involving solutions to differential equations.

Piecewise Functions

A piecewise function is defined by different expressions over different intervals of the domain. The function \(f_1(x)\) in this case is an example. For \(x \geq 0\), the function behaves like \(x^2\), and for \(x < 0\), it acts like \(3x^3\).

Piecewise functions are particularly useful in modeling situations where a rule changes at specific values of \(x\).

To work effectively with piecewise functions, you must consider each segment separately. This includes finding its derivative, calculating integrals, or evaluating expressions. In the example above, \(f_1(x)\) required different treatment when finding its derivative over its respective intervals. Understanding how to handle each piece is vital for correct results, especially in calculus and real-world applications.

Derivative

A derivative represents the rate of change of a function with respect to a variable. In simpler terms, it's the slope of the tangent line to the function at any given point.

For a given function \(y = f(x)\), the derivative denoted as \(f'(x)\) is calculated using differentiation rules. For example:

• The derivative of \(x^n\) is \(n \cdot x^{n-1}\).
• The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.

In our problem, the derivative of \(f_1(x)\) changes based on the piecewise definition, i.e., \(2x\) for \(x \geq 0\) and \(9x^2\) for \(x < 0\).

When working with derivatives, especially of piecewise functions, ensure you find the derivative separately for each piece. This allows for accurate calculations when analyzing properties like continuity, differentiability, and in our case, linear independence.

Interval

An interval in mathematics refers to a range of values. Intervals can be finite, such as \([a, b]\), or infinite, such as \((-\infty, \infty)\).

In the given problem, we evaluate the functions over the entire real line \((-\infty, \infty)\). Checking behavior over different parts of the interval is essential, particularly when dealing with piecewise functions.

Intervals help us to set boundaries for where certain properties must hold true. For instance, when examining the Wronskian in this exercise, it's crucial to check its behavior across varying intervals. This is because piecewise functions might behave differently across those segments. Knowing how to manage and evaluate these segments ensures comprehensive understanding of function behavior and properties like continuity, differentiability, and independence.